Active: Veigar calls a great mass of dark matter to fall from the sky to the target area, which lands after a 1.2 second delay, dealing magic damage. This ability also provides vision through fog of war or brushes before it lands.
Cooldown: 10 seconds
Range: 900
Radius: 225 (estimate)
Cost: 70/80/90/100/110 mana
Magic Damage: 120/170/220/270/320 (+1.0 per ability power)

Starting items against AD mid laner

Health Potion

x3

Limited to 5 at one time.
Click to Consume: Restores 150 Health over 15 seconds.

Cloth Armor

+15 Armor

Core items

Zhonya's Hourglass

+100 Ability Power
+45 Armor
UNIQUE Active - Stasis: Champion becomes invulnerable and untargetable for 2.5 seconds, but is unable to move, attack, cast spells, or use items during this time (90 second cooldown).

Rabadon's Deathcap

+120 Ability Power
Unique: +35% ability power

Endgame items "

This is high burst build with some MR so you will not get nuked by their apc. Unless it's fed.

+100 Ability Power
+45 Armor
UNIQUE Active - Stasis: Champion becomes invulnerable and untargetable for 2.5 seconds, but is unable to move, attack, cast spells, or use items during this time (90 second cooldown).

Rabadon's Deathcap

+120 Ability Power
Unique: +35% ability power

Endgame items against MR and tanks. "

This build provides nice burst with some health and magic penetration. With this you can one shot full built Garen (You will need cca 1100 AP which is not that hard with Veigar :D )

Rylai's Crystal Scepter

+400 Health
+100 Ability Power
UNIQUE Passive: Damaging spells and abilities apply a Movement Speed reduction to enemies based on the spell type:
Single Target: 40% reduction for 1.5 seconds.
Area of Effect: 40% reduction for 1 seconds.
Damage over Time or Multi-hit: 20% reduction for 1 seconds.
Summoned Minions: 20% reduction for 1 seconds.
(If a spell fits in more than one category, it uses the weakest slow value.)

+100 Ability Power
+45 Armor
UNIQUE Active - Stasis: Champion becomes invulnerable and untargetable for 2.5 seconds, but is unable to move, attack, cast spells, or use items during this time (90 second cooldown).

Build variability

This are just two basic builds for Veigar. However, this doesn't cover all posibilities. For example, when playing against your ap counters, you will have to build some magic resist to survive their burst (Fizz,Katarina,LeBlanc,Mordekaiser). Best would be Abyssal Scepter. If you're laning against your ad counter (Talon, Zed), start with 1x cloth armour and build it into Seeker's Armguard. This should provide enough armour to provide good burst reduction and some ability power. Are you surprised that you see lichbane in my build? This build has prooved itself worthy after I have one shot full built Garen. Because in lategame your E+W+DFG+R+Q combo can deal up to 3600 damage. With Lichbane and another 80% ability power autoattack it can be even more. Around 4000 damage. And yes, it will be good if you're against team tanky as fck.

Skillorder

Always max your stun first.

Always try to have your stun maxed, so you can land dark matter on your enemy's head. It is better to have sure damage output than missed big potential and wasted mana. This skill order should provide enough crowd control and burst to kill your enemy. However, don't forget to harras your opponent before trying to nuke him. Try to E+W+AA him, and when he's about 60% of health, you can use your full combo(if you have mana, don't forget to ask for blue your jungler). Just E+W+R+Q+AA+ignite. Why is there that autoattack? Most of the times, when you're not fed yet, you have to do that auto attack or your opponent could escape with cca 20 HP left, and there is nothing more frustrating, than wasting your burst, mana and time to not kill an enemy.

Summoner spells

Spells

Ignite

Flash

Defensive sustain

I find Flash with Ignite as personal favourite. Flash can be used as offensive or defensive spell. E.G. your enemy is about 50% HP but he is out of your range, just Flash towards him, E+W+R+Q+AA and he should die. If you have problem to manage your mana, I recommend you to pick clarity instead of ignite, you will have to harras a little more than with ignite, but you will also have bigger mana pool so you can outsustain, harras and burst your enemy.

Runes

Glymph

x9

Greater Glyph of Potency

+1.19 ability power

Mark

x9

Greater Mark of Potency

+0.59 ability power

Seal

x9

Greater Seal of Potency

+0.59 ability power

Quintessence

I am poor.

I have full Ap runepage just because I can't afford third runepage ( yes, when I have enough IP, I buy a champ). Despite fact, this is full Ap runepage, it provides good starting advantage for you because of Veigar's Q. With this, you can have about 75-110 ability power in 15th minute, which is really good. After 15th minute you will probbably have Needlessly Large rod or DFG itself. Then you will have around 250-300 ability power and you can be bursting your enemy APc all day long.